This paper addresses the computational challenges of the covariance matrix adaptation evolution strategy (CMA-ES) in high-dimensional optimization of an objective function \(f:\mathbb{R}^{n}\to\mathbb{R}\) where \(n\gg 100\) , particularly for problems exhibiting sparse-sensitivity structures where only a small number of directions are highly sensitive. To overcome the bottleneck associated with the covariance matrix, we propose a novel restricted covariance matrix model, reducing storage requirements from quadratic to linear in \(n\) . An efficient adaptation mechanism for this model is introduced, with linear time complexity per iteration, providing a substantial improvement over the standard CMA-ES. We also present two-point step-size adaptation (TPA) as an alternative to the cumulative step-size adaptation (CSA), particularly suitable for high-dimensional optimization. In contrast to CSA, TPA is not negatively affected by ineffective subspaces, prevalent in higher dimensions where non-relevant design variables may be introduced. The proposed algorithm, dd-RCMA-ES, determines its hyper-parameters based on the problem dimension \(n\) and the model complexity \(k\) , thereby maintaining the out-of-the-box usability of the standard CMA-ES, while offering the flexibility to upper bound the model complexity if desired. Experimental evaluations across 18 test problems in dimensions varying from 80 to 10240 demonstrate the efficiency of dd-RCMA-ES in terms of the number of function calls required to achieve a given target precision. In comparison with other CMA-ES variants, the dd-RCMA-ES excels on problems with sparse structures that comprise a small number of both, highly sensitive and insensitive directions.
Evolutionary multitasking (EMT) has emerged as a popular topic of evolutionary computation over the past decade. It aims to concurrently address multiple optimization tasks within limited computing resources, leveraging inter-task knowledge transfer techniques. Despite the abundance of multitask evolutionary algorithms (MTEAs) proposed for multitask optimization (MTO), there remains a need for a comprehensive software platform to help researchers evaluate MTEA performance on benchmark MTO problems as well as explore real-world applications. To bridge this gap, we introduce the first open-source benchmarking platform, named MToP, for EMT. MToP incorporates over 50 MTEAs, more than 200 MTO problem cases with real-world applications, and over 20 performance metrics. Based on these, we provide benchmarking recommendations tailored for different MTO scenarios. Moreover, to facilitate comparative analyses between MTEAs and traditional evolutionary algorithms, we adapted over 50 popular single-task evolutionary algorithms to address MTO problems. Notably, we release extensive pre-run experimental data on benchmark suites to enhance reproducibility and reduce computational overhead for researchers. MToP features a user-friendly graphical interface, facilitating results analysis, data export, and schematic visualization. More importantly, MToP is designed with extensibility in mind, allowing users to develop new algorithms and tackle emerging problem domains. The source code of MToP is available at: https://github.com/intLyc/MTO-Platform
In complex control tasks, finding high-performing policies often requires discovering and exploiting specific behavioral strategies. While Quality-Diversity (QD) algorithms can uncover these strategies through extensive behavior space exploration, they sacrifice efficiency by improving suboptimal behaviors. Conversely, Evolution Strategies (ES) achieve impressive performance through focused optimization but frequently become trapped in local optima due to limited behavioral exploration. We present Quality with J ust E nough Di versity (JEDi), a new optimization framework that resolves this fundamental tension. JEDi employs a combination of Gaussian Process modeling and parallel ES to intelligently explore behavioral space while maintaining focused optimization. At its core, JEDi learns a probabilistic mapping between behaviors and performance, using this model to identify and target promising behavioral regions that could unlock better solutions. This targeted exploration is achieved through multiple Evolution Strategy emitters that simultaneously optimize toward selected behaviors while maximizing task performance. To further improve JEDi’s exploration capabilities, we introduce its Dynamic variant DyJEDi with an adaptive restart mechanism that dynamically detects and responds to emitter convergence, independently restarting each Evolution Strategy when it stagnates in both behavior and fitness space. This dynamic approach significantly improves exploration efficiency and robustness to local optima. We demonstrate that DyJEDi outperforms both traditional ES and QD approaches across challenging continuous robotics control tasks, achieving higher final performance. Most notably, DyJEDi solves several hard exploration problems where standard ES methods consistently fail.
Many existing surrogate-assisted multiobjective evolutionary algorithms (SAMOEAs) struggle in high-dimensional search spaces. Recently, efforts to address this important challenge have intensified, leading to intense performance competition. However, most performance comparisons have been conducted solely among SAMOEAs, with limited benchmarking against standard optimizers such as MOEAs. This lack of a comprehensive comparison has obscured substantive progress in the field and raised concerns regarding the effectiveness of SAMOEAs over MOEAs. To fill this gap, this paper presents a comparative study of SAMOEAs designed for expensive high-dimensional problems. We evaluate eight SAMOEAs against seven MOEAs across six test suites with a limited number of function evaluations, where all algorithms are configured with their default parameter settings. Surprisingly, our results indicate that popular baseline SAMOEAs perform comparably to, or even worse than, most MOEAs. Although state-of-the-art SAMOEAs, particularly decomposition-based algorithms, generally outperform MOEAs, their advantages diminish in certain test suites. Further analysis suggests that these decomposition-based algorithms may be overfitted to the specific shape of the feasible region commonly observed in popular test suites, and this observation was also confirmed in two real-world applications. Based on these findings, we propose an improved experimental design to better assess the effectiveness of SAMOEAs. Moreover, we suggest the importance of developing SAMOEAs that adapt to various shapes of feasible regions, contributing to the further advancement of SAMOEAs.
Vision-based decision-making is relevant to many domains, including safety-critical ones where transparency matters as much as performance. Therefore, automating sequential decision making in such settings requires approaches that balance effectiveness with interpretability. While deep reinforcement learning techniques based on artificial neural networks have achieved strong performance, their black-box nature typically necessitates post hoc explainability analyses. To address this limitation, we propose an approach based on Graph-based Genetic Programming (GGP) that generates policies in the form of computer code, which is fully observable and inherently interpretable. To improve both performance and robustness, we expose GGP-based visual control policies to multiple representative conditions during optimization, mitigating convergence to strong local optima and fragility under changing conditions. Finally, to gain insights into the evolutionary search dynamics, we employ Search Trajectory Networks, an analytical and visualization tool for studying optimization behavior. Our results demonstrate that the resulting policies approach human-level performance and empirically confirm the presence of strong local optima acting as attractors during evolution, providing new insights into the behavior and potential limitations of interpretable evolutionary policy search approaches.
Incorporating multiple elements of evolution strategies, a first-order method for unconstrained optimization that is invariant to strictly increasing function value transformations is proposed. The algorithm subjects normalized gradient vectors to a linear transformation that is adapted based on the directions of successive gradients. In computer experiments, the algorithm is found to often locate near optimal solutions to non-quadratic problems with fewer function evaluations than a quasi-Newton algorithm and a trust-region method.
To address problems with multiple conflicting objective functions while incorporating preference information from a decision maker (a domain expert), interactive evolutionary multiobjective optimization methods have been proposed and widely adopted. To systematically assess the performance of these methods, it is desirable to have indicators that not only rely on the Pareto dominance relation, where solutions with better objective function values are preferred, but also take the decision maker's preferences into account. In this way, indicator values can meaningfully reflect the decision maker's preferences and the quality of the solutions obtained. In this paper, we propose the first performance indicator that incorporates preference information in the form of two reference points, specifically, an aspiration point, consisting of aspiration levels that represent the decision maker's full satisfaction, and a reservation point, consisting of reservation levels that represent dissatisfaction. The new indicator, termed D-PHI, translates information about reservation and aspiration points into a transformed hypervolume measure that assesses the performance of a solution set. We introduce D-PHI with an extended set of guidelines for preference-based performance evaluation, discuss its theoretical properties, and demonstrate its behavior through empirical examples. Moreover, we define a complementary indicator (CI) based on a so-called achievement scalarizing function. CI offers additional insight into whether aspiration and reservation levels have been achieved, and its usefulness is demonstrated through examples.
The Leading Ones Trailing Zeros (LOTZ) benchmark is a well-established bi-objective pseudo-Boolean function (of scalable size) that has both a linear-sized Pareto front, and also a front that is “far away” from a random solution. As such it is a useful function for understanding the behaviour of multi-objective optimization algorithms, especially regarding their ability to approach the front and expand and maintain solutions all along it. It has contributed to both algorithm design and theoretical analysis. In this paper, we make use of three variants of LOTZ, recently introduced , which generalized it to the many-objective case in different ways while aiming to maintain similar properties of the original. We add further detail to the mathematical description of the benchmarks and use them to analyse the performance of a simple multi-objective local search algorithm (PAES-25) (based on (1+1)-PAES) with plug-in modules for archiving, neighbourhood, selection method (from the archive or not), and acceptance function. We measure the anytime performance of PAES-25 in different variants by observing a suite of online measures that give an insight into the dynamics of the algorithm and provide guidance as to choice of configuration depending on factors such as time available and size of Pareto approximation set desired . We believe that the benchmarks and the anytime, multidimensional methods used for analysis might motivate further theory (especially of many-objective optimization and archiving) and may also contribute usefully to algorithm design ideas.
This paper proposes the Augmented Decision Space Optimization (ADSO) framework for sparsity-driven optimization of mixed strategies in Stackelberg Security Games (SSGs). Unlike conventional evolutionary search processes that gradually converge toward sparse solutions, ADSO integrates binary variables to explicitly encode the inclusion or exclusion of pure strategies, while real-valued variables fine-tune their selection probabilities. This dual representation directly enforces sparsity and facilitates computational efficiency in large-scale problem instances. Extensive empirical studies on three benchmark games demonstrate that ADSO consistently produces compact strategies with minimal computational cost, achieving performance comparable to exact methods where they are feasible, and delivering state-of-the-art results in settings beyond the reach of such methods. Apart from SSGs, the framework exhibits strong potential for broader application to other game-theoretic and combinatorial optimization problems characterized by sparse solutions.
How people spend their finite time budget of 24 hours on daily activities is linked to their wellbeing. Yet, how to best allocate time to optimise multi-dimensional wellbeing (physical, mental and cognitive) remains unknown. Here, we utilise a number of (objective) functions derived using compositional data analysis and a large child cohort ( \(n>1{,}000\) ), to predict how time allocation is associated with wellbeing outcomes such as body mass index, life satisfaction and cognition. We develop and advocate joint cumulative distribution function constraints to ensure the feasible solutions do not extrapolate the sampled data for which the objective function is derived from. Moreover, we incorporate quality diversity (QD) approaches to study these objective functions. We define two types of behavioural spaces (BSs), one based on the activities, called the variable-based behavioural space (VBS), and the other based on the objectives, called the objective-based behavioural space (OBS). The VBS allows us to generate a set of high-quality solutions with different activity durations, while the OBS allows us to tradeoff different wellbeing dimensions against each other. We also demonstrate a web application, Time allocation optimiser, for creating personalised, optimised time-use plans.
This is an erratum for the article “Introduction to the Special Issue on Large-Scale Optimization and Learning” published in ACM Trans. Evol. Learn. Optim. 5, 4, Article 23 (December 2025), 3 pages.
The evaluation of heuristic optimizers on test problems, better known as benchmarking, is a cornerstone of research in multi-objective optimization. However, most test problems used in benchmarking numerical multi-objective black-box optimizers come from one of two flawed approaches: On the one hand, problems are constructed manually, which result in problems with well-understood optimal solutions, but unrealistic properties and biases. On the other hand, more realistic and complex single-objective problems are composited into multi-objective problems, but with a lack of control and understanding of problem properties. This paper proposes an extensive problem generation approach for bi-objective numerical optimization problems consisting of the combination of theoretically well-understood convex-quadratic functions into unimodal and multimodal landscapes with and without global structure. It supports configuration of test problem properties, such as the number of decision variables, local optima, Pareto front shape, plateaus in the objective space, or degree of conditioning, while maintaining theoretical tractability: The optimal front can be approximated to an arbitrary degree of precision regarding Pareto-compliant performance indicators such as the hypervolume or the exact R2 indicator. To demonstrate the generator's capabilities, a test suite of 20 problem categories, called BONO-Bench, is created and subsequently used as a basis of an illustrative benchmark study. Finally, the general approach underlying our proposed generator, together with the associated test suite, is publicly released in the Python package to facilitate reproducible benchmarking.
We address the electric autonomous dial-a-ride problem (E-ADARP), a challenging extension of the dial-a-ride problem with the goal of finding minimum cost routes serving given transportation requests with a fleet of electric and autonomous vehicles (EAVs). Special emphasis lies on the minimization of user excess ride time under consideration of the charging requirements of the EAVs while operational constraints have to be satisfied. We propose a novel large neighborhood search (LNS) approach for the E-ADARP together with two alternatives for handling the charging of the EAVs, the scheduling, and route evaluation. One deals with these challenges separately using dedicated LNS operators and a forward labeling algorithm and the other provides a combined approach with on-the-fly charging stop insertion during route evaluation. The performance of the algorithms is evaluated on various configurations of two common benchmark sets as well as some very large-scale instances. Results show that especially the approach with the on-the-fly insertion almost consistently outperforms former state-of-the-art techniques on the common benchmark instances, finding many new best known solutions. For this best performing approach, multiple variants of a more advanced destroy operator for the underlying LNS are investigated. This enhancement can yield significantly improved performance especially on the very large instances as illustrated by further empirical results.
Training-free metrics, also known as zero-cost (ZC) proxies, enable efficient exploration in Neural Architecture Search (NAS) but are less effective than training-based metrics like validation accuracy in identifying high-performance networks. In this article, we investigate the effectiveness of ZC proxies by taking a deeper look into their fitness landscapes utilizing Local Optima Networks. We introduce MF-NAS, a two-stage NAS framework that first employs a ZC proxy-guided local search algorithm to explore the search space. Networks with the highest ZC scores are then fed into the Successive Halving (SH) algorithm to identify the top-performing architecture. However, we observe considerable performance gaps when different ZC metrics are employed. Analyzing those MF-NAS variants with Search Trajectory Networks, we find that high-performance networks could be encountered early or midway through the ZC proxy-guided local search process, making selection based solely on the highest ZC scores ineffective in certain cases. To address this issue, we propose the R-MF-NAS framework, where the selected networks for SH include not only those with the highest ZC scores but also promising solutions encountered during the local search stage. Experiments on diverse NAS benchmarks demonstrate the superiority of both MF-NAS and R-MF-NAS over state-of-the-art methods under a strict budget.
Recent studies have suggested that Large Language Models (LLMs) could provide interesting ideas contributing to mathematical discovery. This claim was motivated by reports that LLM-based genetic algorithms produced heuristics offering new insights into the online bin packing problem under uniform and Weibull distributions. In this work, we reassess this claim through a detailed analysis of the heuristics produced by LLMs, examining both their behavior and interpretability. Despite being human-readable, these heuristics remain largely opaque even to domain experts. Building on this analysis, we propose a new class of algorithms tailored to these specific bin packing instances. The derived algorithms are significantly simpler, more efficient, more interpretable, and more generalizable, suggesting that the considered instances are themselves relatively simple. We then discuss the limitations of the claim regarding LLMs' contribution to this problem, which appears to rest on the mistaken assumption that the instances had previously been studied. Our findings instead emphasize the need for rigorous validation and contextualization when assessing the scientific value of LLM-generated outputs.
Survival analysis is widely utilized to model time-to-event data across biomedical, epidemiological, and engineering domains. However, traditional methods such as Cox regression impose strong assumptions, including proportional hazards, and are often inadequate for capturing complex, non-linear relationships in high-dimensional or heterogeneous data. Rule-based machine learning algorithms, such as “ExSTraCS,” can interpretably model complex biomedical associations in classification tasks. This study extends ExSTraCS to the challenges of right-censored survival data to yield “Survival-LCS,” the first rule-based survival analysis algorithm that additionally handles heterogeneous feature types and missing values and makes no assumptions about baseline hazard or survival time distributions. Survival-LCS is evaluated and compared across a variety of simulated genetic survival datasets covering distinct genetic architectures (additive, epistatic, heterogeneous, and univariate), censoring values, number of features, and survival distributions (random monotonic spline, Gamma, Gaussian, and Weibull) using Integrated Brier Scores and statistical significance testing. Results demonstrate that Survival-LCS reliably captures complex associations with survival outcomes, performing competitively with standard approaches, particularly in settings that challenge traditional models. This work highlights the capability of rule-based algorithms to be adapted as interpretable, assumption-free (i.e., data-driven) survival analysis methods, particularly in domains with higher-dimensional data or complex underlying associations.
In certain real-world optimization scenarios, practitioners are not interested in solving multiple problems but rather in finding the best solution to a single, specific problem. When the computational budget is large relative to the cost of evaluating a candidate solution, multiple heuristic alternatives can be tried to solve the same given problem, each possibly with a different algorithm, parameter configuration, initialization, or stopping criterion. The sequential selection of which alternative to try next is crucial for efficiently identifying the one that provides the best possible solution across multiple attempts. Despite the relevance of this problem in practice, it has not yet been the exclusive focus of any existing review. Several sequential alternative selection strategies have been proposed in different research topics, but they have not been comprehensively and systematically unified under a common perspective. This work presents a focused review of single-problem multi-attempt heuristic optimization. It brings together suitable strategies to this problem that have been studied separately through algorithm selection, parameter tuning, multi-start and resource allocation. These strategies are explained using a unified terminology within a common framework, which supports the development of a taxonomy for systematically organizing and classifying them.
Gradient boosting represents an effective approach for constructing ensembles. We demonstrate how genetic programming can take advantage of the method for a wide range of classification tasks. The resulting Gradient Boosted Programming approach assumes two phases. Phase 1 develops a diverse set of base learners (programs). Phase 2 applies a gradient boosting approach specific to the program representation. The resulting ensemble is additively constructed and a class probability distribution is learnt for each program. An extensive benchmarking study is conducted across 21 classification datasets that include requirements for operation under class imbalance, tens of classes, and feature identification. The proposed approach is significantly better under the 11 low cardinality classification tasks and generally identifies simpler models than other ensemble methods such as Random Forests and XGBoost.
Dynamic Multimodal Optimization Problems (DMMOPs) demand algorithms capable of swiftly locating and tracking multiple optimal solutions over time. The primary challenge lies in controlling the population diversity to facilitate effective exploration, all within the limitation of computational resources between consecutive environmental changes. In this article, we study the utilization of density information derived from both current and historical populations to enhance exploration. First, for each active sub-population, we construct a density landscape based on the distribution of concurrently active sub-populations and establish dominance relationships between candidate solutions in the sub-population based on density and fitness values, directing this sub-population toward exploring low-density promising areas. Then, for each converged sub-population, we construct a density landscape based on the distribution of sub-populations that have historically become extinct, guiding the restart of this sub-population in low-density unexploited areas. Finally, we develop a comprehensive framework of Density-Assisted Evolutionary Algorithm (DAEA), which encompasses densityassisted search and restart, also combined with initialization. Moreover, we employ prediction and memory strategies to enhance the performance of DAEA in dynamic environments. Notably, the algorithm relies on an external monitor to detect environmental changes and trigger the dynamic response strategy. DAEA is tested on the CEC'2022 dynamic multimodal optimization benchmark suite and is compared against several state-of-the-art dynamic multimodal optimization algorithms. The experimental results demonstrate the competitiveness of DAEA in handling DMMOPs. Additionally, experimental results from the berth allocation problem further confirm the applicability of DAEA to real-world dynamic multimodal optimization tasks.
